Optimization Practice Problems

Problem #1Find two nonnegative numbers whose sum is 9 and so the product of one number and the square of the other number is a maximum.

Problem #2Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen?

Problem #3An open rectangular box with a square base is to be made from 48 feet squared of material. What dimensions will result in a box with the largest possible volume?

Problem #4A sheet of cardboard 3 feet by 4 feet will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with the largest volume?

Problem #5A cylindrical can is to hold 20pi m3. The material for the top and bottom costs $10/m2 and the sides cost $8/m2. Find the radius r and height h of the most economical can.

Problem #6Construct a window in the shape of a semi-circle over a rectangle (Norman Window). If the distance around the outside of the window is 12 feet, what dimensions will result in the rectangle having largest possible area?

Problem #7Consider a rectangle of perimeter 12 inches. Form a cylinder by revolving this rectangle about one of its edges. What dimensions of the rectangle will result in a cylinder of maximum volume?

Problem #8A movie screen on a wall is 20 feet high and 10 feet above the floor. At what distance x from the front of the room should you position yourself so that the viewing angle, theta, of the movie screen is as large as possible?

Problem #9Find the dimensions of the rectangle of largest possible area which can be inscribed in the closed region bounded by the x-axis, y-axis, and the graph of y=8-x3.

Problem #10What angle, theta, between two edges of length 3 will result in an isosceles triangle with the largest area?

Answers1. x=3, y =6, product=108

2. X = 50ft, y =125 ft, Area=6250 ft2

3. x=4ft, y = 2ft, V=32 ft3

4. x= 0.57 ft, V=3.03 ft3

5. R=2m, h=5m, C=$754

6. X=1.17ft, 2x=2.34 ft, y = 3ft, a = 6.99ft2

7. r=4ft, h =2ft, v=100.53ft3

8. Theta= pi/6=30

9. X=1.26, y=6, A=7.6

10. Theta=pi/2=90, a=9/2